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Iván Angiono

Publications and source records attributed to Iván Angiono.

At least 37 records · Page 2Linked to original sources
arXiv2020

On Nichols algebras over basic Hopf algebras

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field $\Bbbk$ of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter-Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand-Kirillov dimension.

math.QA↗
arXiv2019

Examples of finite-dimensional pointed Hopf algebras in positive characteristic

We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension described in arXiv:1606.02521. We obtain nex examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

math.QA↗
arXiv2019

Pointed Hopf algebras over non abelian groups with decomposable braidings, I

We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.

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arXiv2018

Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type $V$ with a principal realization in the category of Yetter-Drinfeld modules of a cosemisimple Hopf algebra $H$ and such that the Nichols algebra $\mathfrak{B}(V)$ is finite-dimensional, thus presented by a finite set $\mathcal G$ of relations, we define a family of Hopf algebras $\mathfrak{u}(\boldsymbolλ)$, $\boldsymbolλ\in \Bbbk^{\mathcal G}$, which are cocycle deformations of $\mathfrak{B}(V)\# H$ and such that $\operatorname{gr}\mathfrak{u}(\boldsymbolλ)\simeq \mathfrak{B}(V)\# H$.

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arXiv2018

Pointed Hopf algebras: a guided tour to the liftings

This article serves a two-fold purpose. On the one hand, it is a survey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of type $B_2$.

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arXiv2018

On finite GK-dimensional Nichols algebras over abelian groups

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname{GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb Z$ with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite $\operatorname{GKdim}$ if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite $\operatorname{GKdim}$ if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite $\operatorname{GKdim}$. Consequently we present several new examples of Nichols algebras with finite $\operatorname{GKdim}$, including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

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arXiv2018

On finite GK-dimensional Nichols algebras of diagonal type

It was conjectured in \texttt{\small arXiv:1606.02521} that a Nichols algebra of diagonal type with finite Gelfand-Kirillov dimension has finite (generalized) root system. We prove the conjecture assuming that the rank is 2. We also show that a Nichols algebra of affine Cartan type has infinite Gelfand-Kirillov dimension.

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arXiv2017

On Finite dimensional Nichols algebras of diagonal type

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in [H-classif RS] as a notable application of the notions of Weyl groupoid and generalized root system [H-Weyl gpd,HY]. In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of [H-classif RS] the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in [A-jems,A-presentation]; the PBW-basis; the dimension or the Gelfand-Kirillov dimension; the associated Lie algebra as in [AAR2]. Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.

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arXiv2017

Pointed finite tensor categories over abelian groups

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group.

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arXiv2017

A quantum version of the algebra of distributions of $\operatorname{SL}_2$

Let $λ$ be a primitive root of unity of order $\ell$. We introduce a family of finite-dimensional algebras $\{\mathcal{D}_{λ,N}(\mathfrak{sl}_2)\}_{N\in\mathbb{N}_0}$ over the complex numbers, such that $\mathcal{D}_{λ,N}(\mathfrak{sl}_2)$ is a subalgebra of $\mathcal{D}_{λ,M}(\mathfrak{sl}_2)$ if $N<M$, and $\mathcal{D}_{λ,N-1}(\mathfrak{sl}_2)\subset \mathcal{D}_{λ,N}(\mathfrak{sl}_2)$ is a $\mathfrak{u}_λ(\mathfrak{sl}_2)$-cleft extension. The simple $\mathcal{D}_{λ,N}(\mathfrak{sl}_2)$-modules $(\mathcal{L}_{N}(p))_{0\le p<\ell^{N+1}}$ are highest weight modules, which admit a tensor product decomposition: the first factor is a simple $\mathfrak{u}_λ(\mathfrak{sl}_2)$-module and the second factor is a simple $\mathcal{D}_{λ,N-1}(\mathfrak{sl}_2)$-module. This factorization resembles the corresponding Steinberg decomposition, and the family of algebras resembles the presentation of algebra of distributions of $\operatorname{SL}_2$ as a filtration by finite-dimensional subalgebras.

math.RA↗
arXiv2016

The quantum divided power algebra of a finite-dimensional Nichols algebra of diagonal type

Let $\mathcal{B}_\mathfrak{q}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q}$. We consider the graded dual $\mathcal{L}_{\mathfrak{q}}$ of the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}_{\mathfrak{q}}$ from [A3] and the divided powers algebra $\mathcal{U}_{\mathfrak{q}}$, a suitable Drinfeld double of $\mathcal{L}_{\mathfrak{q}} # \mathbf{k} \mathbb{Z}^θ$. We provide basis and presentations by generators and relations of $\mathcal{L}_{\mathfrak{q}}$ and $\mathcal{U}_{\mathfrak{q}}$, and prove that they are noetherian and have finite Gelfand-Kirillov dimension.

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arXiv2016

Liftings of Jordan and super Jordan planes

We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

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arXiv2016

Liftings of Nichols algebras of diagonal type I. Cartan type A

After the classification of the finite-dimensional Nichols algebras of diagonal type arXiv:math/0411477, arXiv:math/0605795, the determination of its defining relations arXiv:1008.4144, arXiv:1104.0268, and the verification of the generation in degree one conjecture arXiv:1104.0268, there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question was developed in arXiv:1212.5279, built on cocycle deformations. In this paper, we elaborate further and present an explicit algorithm to compute liftings. In our main result we classify all liftings of finite-dimensional Nichols algebras of Cartan type $A$, over a cosemisimple Hopf algebra $H$. This extends arXiv:math/0110136, where it was assumed that the parameter is a root of unity of order $>3$ and that $H$ is a commmutative group algebra. When the parameter is a root of unity of order 2 or 3, new phenomena appear: the quantum Serre relations can be deformed, this allows in turn the power root vectors to be deformed to elements in lower terms of the coradical filtration, but not necessarily in the group algebra. These phenomena are already present in the calculation of the liftings in type $A_2$ at a parameter of order 2 or 3 over an abelian group arXiv:math/0204075, arXiv:1003.5882, done by a different method using a computer program. As a by-product of our calculations, we present new infinite families of finite-dimensional pointed Hopf algebras.

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arXiv2016

A finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{θ\times θ}$, where $\mathbf{k}$ is an algebraically closed field of characteristic 0. Let $\mathcal{L}_{\mathfrak{q}}$ be the Lusztig algebra associated to $\mathcal{B}_{\mathfrak{q}}$, see http://arxiv.org/abs/1501.04518. We present $\mathcal{L}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of $\mathcal{B}_{\mathfrak{q}}$ by $\mathfrak Z_{\mathfrak{q}}$ where $\mathfrak Z_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra of a Lie algebra $\mathfrak n_{\mathfrak{q}}$. We compute the Lie algebra $\mathfrak n_{\mathfrak{q}}$ when $θ= 2$.

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arXiv2015

On rigidity of Nichols algebras

We study deformations of graded braided bialgebras using cohomological methods. In particular, we show that many examples of Nichols algebras, including the finite-dimensional ones arising in the Andruskiewitsch-Schneider program of classification of pointed Hopf algebras, are rigid. This result can be regarded as nonexistence of "braided Lie algebras" with nontrivial bracket.

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arXiv2013

Lifting via cocycle deformation

We develop a strategy to compute all liftings of a Nichols algebra over a finite dimensional cosemisimple Hopf algebra. We produce them as cocycle deformations of the bosonization of these two. In parallel, we study the shape of any such lifting.

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